"F" Button in Shape Key Editor Not Showing Up - Why Not?
The "F" button is to tell Blender to save a data block even if it has no user, so it appears only when... you have a data block!



In your picture, you have this situation for that eye:



the " + new " button means, there's no data block, and you can add one (an action in that case). Then, the "F" button will appear.

edit

I think that eye has something different than the other...



selecting that green NLA editor stripe, and using TAB it expands, and the " + add " becomes usable, but acts weirdly, at least...



I'm not an expert though, and can't understand how to fix it. But maybe you know what happened to that strip?